A242434 - OEIS

%I #15 Jul 15 2020 15:14:29

%S 1,1,0,0,1,3,0,0,0,1,4,0,0,10,60,0,1,5,0,0,15,105,0,0,0,36,286,0,0,

%T 1281,12768,0,0,0,56,504,1,7,2520,27720,28,378,1260,0,0,7014,84000,0,

%U 0,4621,83168,360360,210,2346,2522880,37837800,13860,180180,120,1320

%N Number of compositions of n in which each part p has multiplicity p.

%C a(n) = 0 for n in {A001422}, a(n) > 0 for n in {A003995}.

%H Alois P. Heinz, <a href="/A242434/b242434.txt">Table of n, a(n) for n = 0..4500</a>

%e a(0) = 1: the empty composition.

%e a(1) = 1: [1].

%e a(4) = 1: [2,2].

%e a(5) = 3: [1,2,2], [2,1,2], [2,2,1].

%e a(9) = 1: [3,3,3].

%e a(10) = 4: [1,3,3,3], [3,1,3,3], [3,3,1,3], [3,3,3,1].

%e a(13) = 10: [2,2,3,3,3], [2,3,2,3,3], [2,3,3,2,3], [2,3,3,3,2], [3,2,2,3,3], [3,2,3,2,3], [3,2,3,3,2], [3,3,2,2,3], [3,3,2,3,2], [3,3,3,2,2].

%p b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<1, 0,

%p        b(n, i-1, p) +`if`(i^2>n, 0, b(n-i^2, i-1, p+i)/i!)))

%p     end:

%p a:= n-> b(n, isqrt(n), 0):

%p seq(a(n), n=0..100);

%t b[n_, i_, p_] := b[n, i, p] = If[n==0, p!, If[i<1, 0, b[n, i-1, p] + If[i^2 >n, 0, b[n-i^2, i-1, p+i]/i!]]]; a[n_] := b[n, Floor[Sqrt[n]], 0]; Table[ a[n], {n, 0, 100}] (* _Jean-François Alcover_, Feb 08 2017, translated from Maple *)

%Y Cf. A033461 (the same for partitions), A336269.

%K nonn,look

%O 0,6

%A _Alois P. Heinz_, May 14 2014